Coinductive Predicates and Final Sequences in a Fibration Ichiro - PowerPoint PPT Presentation

Establish/Compute/Construct Coinductive Predicates ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X � � � X · · · � 0 � � 1 � � 2 � Hasuo (Tokyo)

Establish/Compute/Construct Coinductive Predicates ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X � � � X · · · � | � n � , but � | = ν u. � u = n � 0 � � 1 � � 2 � Hasuo (Tokyo)

Establish/Compute/Construct Coinductive Predicates ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X � � � X · · · � | � n � , but � | = ν u. � u = n � 0 � � 1 � � 2 � State space bound [Cousot & Cousot, ’79] | X | steps Hasuo (Tokyo)

Establish/Compute/Construct Coinductive Predicates ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X � � � X · · · � | � n � , but � | = ν u. � u = n � 0 � � 1 � � 2 � State space bound [Cousot & Cousot, ’79] | X | steps “Behavioral bound” [Hennessy & Milner, ’85] ω steps if finitely branching ! Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. Proof : Suffices to show is an invariant. � � n � n Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x Proof : Suffices to show ⋯ is an invariant. � x 1 x 2 x k � n � n Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 0 � Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 1 � � 0 � Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 1 � � 0 � � 2 � Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 1 � � 0 � � 3 � � 2 � Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 1 � � 0 � � 3 � � 2 � ⋮ ⋮ ⋮ Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred. P ( X ) Theorem. Let a Kripke frame be finitely branching. Then c � X ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · stabilizes after ω steps. x � � n � Proof : Suffices to show n ⋯ is an invariant. � x 1 x 2 x k � n � n � 1 � � 0 � � 3 � = � n � � i � [1 , k ] s.t. x i | � 2 � ⋮ ⋮ for infinitely many n ⋮ Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P ?? � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P ?? � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X By Knaster-Tarski Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P ?? � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X By Knaster-Tarski Inductive constr. ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P ?? � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X By Knaster-Tarski Inductive constr. ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · State space bound vs. “ behavioral bound ” Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary) (current st.) ⊨ P witnesses (next st.) ⊨ P ?? � � c − 1 ϕ � 2 X � 2 P X � 2 X � ν u. � u � = gfp P ( X ) c � X By Knaster-Tarski Inductive constr. ( c − 1 � ϕ � ) X ( c − 1 � ϕ � ) 2 X X � � � · · · current work State space bound vs. “ behavioral bound ” Hasuo (Tokyo) current work

Part II: Coinductive Predicates, Categorically

Contributions Sufficient condition for categorical behavioral ω -bound based on F X P c � Coalgebra (transition system) ↓ p X C Fibration (underlying logic) ϕ P P � � p Predicate lifting (modality) p � � � � C C � � Y F � Locally presentable category (“size”) . X � m . i i ∃ l o C i . � � � � X ∃ i . . . Hasuo (Tokyo)

Contributions Sufficient condition for categorical behavioral ω -bound based on F X P c � Coalgebra (transition system) ↓ p X C Fibration (underlying logic) ϕ P P � � p Predicate lifting (modality) p � � � � C C � � Y F � Locally presentable category (“size”) . X � m . i i ∃ l o C i . � � � � X ∃ i . . . Constr. of final coalg. by final sequence [Worrell, Adamek] Hasuo (Tokyo)

Contributions Sufficient condition for categorical behavioral ω -bound based on F X P c � Coalgebra (transition system) ↓ p X C Fibration (underlying logic) ϕ P P � � p Predicate lifting (modality) p � � � � C C � � Y F � Locally presentable category (“size”) . X � m . i i ∃ l o C i . � � � � X ∃ i . . . Constr. of final coalg. by Coind. predicate as a final sequence [Worrell, Adamek] final coalgebra [Hermida, Jacobs] Hasuo (Tokyo)

Categorical infrastructure: Contributions fibration and locally presentable cat. Sufficient condition for categorical behavioral ω -bound based on F X P c � Coalgebra (transition system) ↓ p X C Fibration (underlying logic) ϕ P P � � p Predicate lifting (modality) p � � � � C C � � Y F � Locally presentable category (“size”) . X � m . i i ∃ l o C i . � � � � X ∃ i . . . Constr. of final coalg. by Coind. predicate as a final sequence [Worrell, Adamek] final coalgebra [Hermida, Jacobs] Hasuo (Tokyo)

Categorical infrastructure: Some math work Contributions fibration and locally presentable cat. Sufficient condition for categorical behavioral ω -bound based on F X P c � Coalgebra (transition system) ↓ p X C Fibration (underlying logic) ϕ P P � � p Predicate lifting (modality) p � � � � C C � � Y F � Locally presentable category (“size”) . X � m . i i ∃ l o C i . � � � � X ∃ i . . . Constr. of final coalg. by Coind. predicate as a final sequence [Worrell, Adamek] final coalgebra [Hermida, Jacobs] Hasuo (Tokyo)

The Categorical Setup P ω ( X ) Kripke model c O X coinductive ν u. � u specification ϕ 3 : 2 X − → 2 P ω X { X � � X | U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive ν u. � u specification ϕ 3 : 2 X − → 2 P ω X { X � � X | U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup finitely branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive ν u. � u specification ϕ 3 : 2 X − → 2 P ω X { X � � X | U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup finitely finitary branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive ν u. � u specification ϕ 3 : 2 X − → 2 P ω X { X � � X | U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup finitely finitary branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X { X � � X | U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup finitely finitary branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X ϕ 3 2 X monotone ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

The Categorical Setup finitely finitary branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X c ∗ ϕ X / P F X ϕ 3 2 X P X / P X monotone endofunctor ( c − 1 � ϕ 3 ) U ✓ invariant U Hasuo (Tokyo)

O The Categorical Setup finitely finitary branching P ω ( X ) F X Kripke model coalgebra c O c O X X coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X c ∗ ϕ X / P F X ϕ 3 2 X P X / P X monotone endofunctor ( c − 1 � ϕ 3 ) U ( c ∗ � ϕ ) P coalgebra ✓ invariant (in a fibr.) U P Hasuo (Tokyo)

X O coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X c ∗ ϕ X / P F X ϕ 3 2 X P X / P X monotone endofunctor ( c − 1 � ϕ 3 ) U ( c ∗ � ϕ ) P coalgebra ✓ invariant (in a fibr.) U P ( c − 1 � ϕ 3 ) J ν u. ϕ 3 u K c coind. pred. J ν u. ϕ 3 u K c X ◆ ( c − 1 � ϕ 3 ) X ◆ · · · inductive constr. Hasuo (Tokyo)

X O coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X c ∗ ϕ X / P F X ϕ 3 2 X P X / P X monotone endofunctor ( c − 1 � ϕ 3 ) U ( c ∗ � ϕ ) P coalgebra ✓ invariant (in a fibr.) U P ( c − 1 � ϕ 3 ) J ν u. ϕ 3 u K c ( c ∗ � ϕ ) J νϕ K c final coalg. ⇠ = O coind. pred. (in a fibr.) J ν u. ϕ 3 u K c J νϕ K c X ◆ ( c − 1 � ϕ 3 ) X ◆ · · · inductive constr. Hasuo (Tokyo)

X O coinductive coinductive ν u. � u νϕ specification specification ϕ 3 : 2 X − → 2 P ω X ϕ : P X − → P F X { X � � X | predicate lifting U �� X � � U � = � } / 2 P X c − 1 / 2 X c ∗ ϕ X / P F X ϕ 3 2 X P X / P X monotone endofunctor ( c − 1 � ϕ 3 ) U ( c ∗ � ϕ ) P coalgebra ✓ invariant (in a fibr.) U P ( c − 1 � ϕ 3 ) J ν u. ϕ 3 u K c ( c ∗ � ϕ ) J νϕ K c final coalg. ⇠ = O coind. pred. (in a fibr.) J ν u. ϕ 3 u K c J νϕ K c X ◆ ( c − 1 � ϕ 3 ) X ◆ · · · > X ( c ∗ � ϕ X ) > X · · · inductive constr. final sequence in a fibr. Hasuo (Tokyo)

What Categorical Generalization Buys Us Hasuo (Tokyo)

What Categorical Generalization Buys Us Final coalgebra in C : (strongly) LFP ( Posets , Graphs , Vec , ...) [Adamek ’03] Coinductive pred. for different F : Sets → Sets Coalg. μ -calculus ; coalg. automata [Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06] Hasuo (Tokyo)

What Categorical Generalization Buys Us Final coalgebra in C : (strongly) LFP ( Posets , Graphs , Vec , ...) [Adamek ’03] Coinductive pred. for different F : Sets → Sets Coalg. μ -calculus ; coalg. automata [Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06] P Various “underlying logics” as ↓ p C Sub( C ) Rel ↓ Sub(Sets F ) ( C : a topos) ↓ C ↓ Sets Constructive logics Sets F Relations (“binary pred. ”) For name-passing Hasuo (Tokyo)

What Categorical Generalization Buys Us Final coalgebra in C : (strongly) LFP ( Posets , Graphs , Vec , ...) [Adamek ’03] Coinductive pred. for different F : Sets → Sets Coalg. μ -calculus ; coalg. automata [Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06] P Coind. relations Various “underlying logics” as ↓ p e.g. bisimilarity C Sub( C ) Rel ↓ Sub(Sets F ) ( C : a topos) ↓ C ↓ Sets Constructive logics Sets F Relations (“binary pred. ”) For name-passing Hasuo (Tokyo)

Coindution in a Fibration P Rel Pred conventional relational fibrational ↓ p ↓ ↓ C Sets Sets invariant bisimulation coalgebra coind. pred. bisimilarity final coalg. partition inductive constr. final sequence refinement Hasuo (Tokyo)

Part III: Technical Ingredients Final Sequence, Fibration, Predicate Lifting, Locally Finitely Presentable Category, ...

[Worrell, TCS’05] in Sets o o o o [Adamek, TCS’03] in strongly LFP C Final Sequence F i 1 1 F 1 ! · · · · · · Hasuo (Tokyo)

[Worrell, TCS’05] in Sets o o o o [Adamek, TCS’03] in strongly LFP C Final Sequence { i-step behaviors} F i 1 1 F 1 ! · · · · · · Hasuo (Tokyo)

[Worrell, TCS’05] in Sets o s t { o o o [Adamek, TCS’03] in strongly LFP C Final Sequence lim { i-step behaviors} F ω 1 π i F i 1 1 F 1 ! · · · · · · Hasuo (Tokyo)

[Worrell, TCS’05] in Sets k d s s o y _ j o o � o [Adamek, TCS’03] in strongly LFP C Final Sequence lim { i-step behaviors} F ω 1 5 π i 1 F i 1 F 1 ! b · · · · · · F π i − 1 F ( F ω 1) Hasuo (Tokyo)

[Worrell, TCS’05] in Sets o k j o o o d _ y s s � [Adamek, TCS’03] in strongly LFP C Final Sequence lim { i-step behaviors} F ω 1 5 π i 1 F i 1 F 1 ! b · · · · · · F π i − 1 F ( F ω 1) : a final coalgebra? F ω 1 Yes, when F is limit preserving ( b is iso) Hasuo (Tokyo)

[Worrell, TCS’05] in Sets _ s o o k j y o s � d o [Adamek, TCS’03] in strongly LFP C Final Sequence lim { i-step behaviors} F ω 1 5 π i 1 F i 1 F 1 ! b · · · · · · F π i − 1 F ( F ω 1) : a final coalgebra? F ω 1 Yes, when F is limit preserving ( b is iso) Almost, when F is finitary ( b is monic) Quotient modulo beh. eq. Continue till ω + ω [Worrell] Hasuo (Tokyo)

P Fibration ↓ p C f / Y in C X “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

P Fibration ↓ p C f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

P Fibration ↓ p P X C P ’ P ’’ P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

P Fibration ↓ p P X P Y C Q ’ P ’ P ’’ Q P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

P Fibration ↓ p P X P Y C Q ’ P ’ P ’’ indexed entities Q P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

o P Fibration ↓ p P X P Y C Q ’ P ’ f ∗ P ’’ indexed entities Q P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

o P Fibration ↓ p “substitution” P X P Y C Q ’ P ’ f ∗ P ’’ indexed entities Q P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

o P Fibration ↓ p “substitution” P X P Y C Q ’ P ’ f ∗ P ’’ indexed entities Q P f / Y in C X indices “Organize indexed entities,” categorically In particular: categorical model of predicate logics Hasuo (Tokyo)

o o P Fibration ↓ p “substitution” P X P Y C Q ’ P ’ f ∗ P ’’ indexed entities Q P f / Y in C X indices “Organize indexed entities,” categorically : predicates over X ( P X, ⊆ ) In particular: Substitution f − 1 ( V ⊆ Y ) categorical model of f − 1 P X P Y � � = V f ( ) predicate logics f / Y X Hasuo (Tokyo)

o Fibration: from Pointwise Indexing to Display Indexing P X P Y Q ’ f ∗ P ’ P ’’ Q P f / Y X in C Hasuo (Tokyo)